Logarithmic order and type of indeterminate moment problems II

This paper deals with the indeterminate moment problem on the real line. We are given a positive measure μ on R having moments of all orders and we assume that μ is not determined by its moments. For details about the indeterminate moment problem see the monographs by Akhiezer, by Shohat and Tamarkin or the survey paper by Berg. Our notation follows that of Akhiezer. In this indeterminate situation the solutions ν to the moment problem form an infinite convex set V , which is compact in the vague topology. Nevanlinna has obtained a parametrization of V in terms of the so-called Pick functions. We recall that a holomorphic function φ defined in the upper half plane is called a Pick function if =φ(z) ≥ 0 for =z > 0. The class of Pick functions is denoted by P. The Nevanlinna parametrization is the one-to-one correspondence νφ ↔ φ between V and P ∪ {∞} given by ∫ ∞